MULTIMODAL SEMIOTICS OF MATHEMATICS TEACHING AND LEARNING A Dissertation submitted to the Faculty of the Graduate School of Arts and Sciences of Georgetown University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Linguistics By Daniel Ginsberg, M.S., M.A.T. Washington, DC October 21, 2015 Copyright 2015 by Daniel Ginsberg Licensed under the Creative Commons Attribution ShareAlike 4.0 License, http://creativecommons.org/licenses/by-sa/4.0/ ii MULTIMODAL SEMIOTICS OF MATHEMATICS TEACHING AND LEARNING Daniel Ginsberg, M.S., M.A.T. Thesis Advisor:ABSTRACT Mark A. Sicoli, Ph.D. The practice of mathematics education is fundamentally multimodal. It incorporates not only talk and embodied action, but also technical notation and diagrams, brought into discourse through verbal and gestural reference. As this interplay of semiotic systems arises in interaction, it can be interpreted by analyzing sequences of talk, writing, and gesture, but a better understanding requires an ethnographic perspective to contextualize interaction with reference to its physical surroundings, institutional setting, and enduring relationships within the community. Thus, classroom interaction is best understood as a multimodal ecology in which micro-level discursive practices, the history of a community, and the biographies of its members mutually influence and determine one another. Beginning from the ecological perspective on classroom interaction, this dissertation presents an analysis of observations and video recordings collected during a semester’s multi-site ethnographic fieldwork with both a middle school math class for English learners and a quasi- remedial college calculus section. To model how any perceptible feature of the environment may be taken as meaningful, I draw on the semiotic theory of C. S. Peirce, not only in interpreting observational data, but as an organizing principle, as the analysis moves from qualities, to particular instances, to recurring patterns. First, I consider the ontological status of mathematical notation as a quality of interaction, investigating its capacity for representing continuous phenomena. Second, I take up actually occurring sequences of interaction, showing how students’ participation in conversational repair offers insights into ideologies of classroom authority and mathematical knowledge. Third, I address students’ and teachers’ identities in the classroom as social perceptions that are constructed and recognized through patterns of interaction. Each area of iii inquiry is then reconsidered to identify semiotic affordances that are made available in classroom interaction, and to explicate problems of practice that prevent students from seeing themselves as successful mathematics learners. The problems I identify are similar to those addressed by mathematics education researchers in the learning sciences, so I conclude by proposing a future research trajectory combining linguistic anthropology, the learning sciences, and classroom teaching practice. iv ACKNOWLEDGEMENTS As a researcher, I owe an immeasurable debt to many people, but my opportunities to acknowledge them come far too seldom. At this particular moment in my professional career, I am glad to have the chance to name a few of the people without whom this project would not have been possible. Thanks first of all to my participants, Ms. M and Dr. C and all their students. Everything in this dissertation, I owe to them, and I have so little to offer in exchange for their patience and generous hospitality. Thanks to my advisor, Mark Sicoli. His approach to understanding the social world through Languageanthropology as Social and semiot Actionics is clearly present in these pages, as are most of the readings from his seminar. No less important, but hopefully made invisible in this write-up, are all the advising sessions in which he guided me through unfamiliar terrain: the practice of research and the institution of academia. Discourse Analysis: Conversation,Thanks to my committee. When I took Heidi Hamilton’s class I felt like I was being adopted into the scholarly family of interactional sociolinguists, and she has continued to take a hand in my development as a researcher, contributing good sense as well as methodological rigor. Jim Sandefur combines video analysis and math pedagogy in ways that I find inspirational in their very practicality, and his involvement in this project was an early vote of confidence that I could make something of interest to math educators. Fred Erickson has spent a long and brilliant career investigating many of the topics I consider here, from intersubjectivity to multimodality to classroom interaction, and I’m grateful to have had the benefit of his experience. Thanks to all the other linguists who have mentored me in various ways. Anna Trester’s ethnography class was like a religious experience for me, an introduction to a profoundly different way of being in the world. Deborah Tannen taught me that deep thinking should not come at the v expense of clear writing. Maya Honda and Wayne O’Neil inspired me to engage seriously in conversations between teachers and researchers. Barbara Clark has shown me that applied linguistic anthropology can have an audience outside the academy, if you put in the work to find it. Thanks to Erin Esch Pereira, our graduate program coordinator, who translates the institutional—so effortlessly, it seems—into terms I can understand and act on. Without her, this might be a research report, but it wouldn’t be a doctoral dissertation. Thanks to the graduate students of the Georgetown Department of Linguistics, a dynamic and supportive community of scholars. In particular, I want to mention my writing partners, Mackenzie Price and Marta Baffy; my colleagues in the Interaction Analysis Lab, Sylvia Sierra, Jeremy Wegner, Joshua Kraut, Nazir Harb, Aisulu Raspayeva, Didem Ikizoglu, and Adrienne Isaac; and my “older brothers and sisters,” Drs. Amelia Tseng, Anastasia Nylund, Jessi Grieser, Jinsok Lee, Julie Lake, Kaitlyn Tagarelli, Laura Siebecker, and Marissa Fond, who by their example showed me that this was achievable. At the same time, I became a member of a parallel community of “Twitter linguists;” particular thanks are due to Michael Maune, Angus Grieve-Smith, and Jodie Martin. Thanks to all the learning scientists and mathematics educators who have expressed an interest in this work and given suggestions at one time or another: Andy Elby, Ayush Gupta, Chandra Turpen, Joe Redish, and Brian Danielak of the University of Maryland Physics Education Research Group; Richard Barwell, my fellow AAAL mathematician; the Math Twitter Blogosphere, in particular Justin Lanier, Christopher Danielson, Bryan Meyer, and especially Ilana Horn; the International Society for the Learning Sciences, who funded my attendance at the 2014 conference; and Anna Sfard, Paul Cobb, and Judit Moschkovich, who took the time to talk with me while I was there. The last section of the last chapter is for all of them. Thanks to my erstwhile colleagues at the Center for Applied Linguistics, where I learned to think of myself as a researcher. Dorry Kenyon, although he is a quantitative researcher and will be the first to tell you that he’s not a linguist, was the first to point me in the direction of classroom vi discourse. Jim Bauman, Laura Wright, and David MacGregor engaged with me in serious conversations about research before I had the first clue what I was doing. My supervisors, Stephanie Gibson and Jennifer Norton, made space for me to dip my toes into research within the scope of our project work. Thanks to all my students, especially my sheltered-immersion mathematics students at Malden High School, 2007–2009. My desire to understand the relationship between language and mathematics first began in the hope that it would help me teach them better. Thanks to my undergraduate mentor, Jane Hale, my first example of the kind of socially engaged scholar that I aspire to be. Her suggestion that I put off Ph.D. school turned out to be absolutely essential. Thanks to my family. Fredric and Nancy Ginsberg taught me to think in a disciplined way, to work and play with language in all its forms, and to take pleasure in discovery. Rose and Hannah Ginsberg led me to think about teaching as performance, and research as creative art. Meaghan O’Connor was my patient first audience for every half-formed idea, and whenever I let other things go so that I could focus on this project, she was always there to pick up the slack. And finally, Samuel O’Connor Ginsberg, my other great project of the last five years—this is dedicated to you. vii TABLE OF CONTENTS Chapter 1 Finding Meaning in Mathematics............................................................................................ 1 1.1. Mathematics as Communication........................................................................................................ 1 1.2. What a Sociocultural Linguist Sees in a Mathematics Lesson ............................................... 3 1.2.1. Fifty out of a Hundred....................................................................................................................... 4 1.2.2. Observations, Reflections, and Areas of Inquiry.................................................................... 5 1.3. Theory and Method................................................................................................................................. 9 1.3.1. Ethnographic Perspectives in the